Throughout, \(P :=(V, \le )\)  denotes an ordered set (poset), that is
a set \(V\)  equipped with a binary relation \(\le \)  on \(V\)  which is
reflexive, antisymmetric and transitive. We say that two elements \(x,y\in V\)  are comparable if \(x\le y\)  or \(y\le x\) , otherwise, we say they are incomparable. The dual of \(P\)  denoted \(P^{*}\)  is the order defined on \(V\)  as follows: if \(x,y\in V\) , then \(x\le y\)  in \(P^{*}\)  if and only if \(y\le x\)  in \(P\) . Let \(P :=(V, \le )\)  be a poset. A set of pairwise comparable elements is called a chain. On the other hand, a set of pairwise incomparable elements is called an antichain. The width of a poset is the maximum cardinality of its antichains (if the maximum does not exist, the width is set to be infinite). Dilworth's celebrated theorem on finite posets [1]} states that the maximum cardinality of an antichain in a finite poset equals the minimum number of chains needed to cover the poset. This result remains true even if the poset is infinite but has finite width. If the poset \(P\)  has width 2 and the incomparability graph of \(P\)  is connected, the partition of \(P\)  into two chains is unique (picking any vertex \(x\) , observe that the set of vertices at odd distance from \(x\)  and the set of vertices at even distance from \(x\)  form a partition into two chains). This paper is essentially about those posets that we call coverable by two chains instead of having width two. According to Szpilrajn [2]}, every order on a set has a linear extension. Let \(P:=(V,\le )\)  be a poset. A realizer of \(P\)  is a family \(\mathcal {L}\)  of linear extensions of the order of \(P\)  whose intersection is the order of \(P\) . Observe that the set of all linear extensions of \(P\)  is a realizer of \(P\) . The dimension of \(P\) , denoted \(dim(P)\) , is the least cardinal \(d\)  for which there exists a realizer of cardinality \(d\)  [3]}. It follows from the Compactness Theorem of First Order Logic that an order is intersection of at most \(n\)  linear orders (\(n\in {\mathbb {N}}\) ) iff every finite restriction of the order has this property. Hence the class of posets with dimension at most \(n\)  is determined by a set of finite obstructions, each obstruction is a poset \(Q\)  of dimension \(n+1\)  such that the deletion of any element of \(Q\)  leaves a poset of dimension \(n\) ; such a poset is said critical. For \(n\ge 2\)  there are infinitely many critical posets of dimension \(n+1\) . For \(n=2\)  they have been described by Kelly [4]}; beyond, the task is considered as hopeless.
